Scheme for Secure Communication via Information Hiding Based on Key Exchange and Decomposition Protocols

This paper considers a decomposition framework as a mechanism for information hiding for secure communication via open network channels. Two varieties of this framework are provided: one is based on Gaussian arithmetic with complex modulus and another on an elliptic curve modular equation. The proposed algorithm is illustrated in a numerical example.


Introduction and Problem Definition
In this paper it is demonstrated how to use various Diffie-Hellman key establishment (DHKE) protocols in order to design a computationally efficient cryptographic schemes for secure communication between two parties {called Alice and Bob}.One of these key establishment protocols is based on modular elliptic curves (ECDHKE) [1].Another DHKE protocol is based on arithmetic of complex integers with complex modulus [2].
DHKE protocol based on complex integers: In this scheme both parties agree to select a Gaussian integer L = (g, h) := g + ih called generator and a complex modulus (l, p) := l + ip with real integer components l and p. Alice and Bob independently select secret real integers alpha and beta respectively.Alice and Bob respectively compute their public keys: (1) and Then Alice and Bob compute respectively that is a point on (6).The scheme is analogous to (1)-( 5): Alice and Bob select a point Q with high order on (6) and real integers alpha (Alice's secret key) and beta (Bob's secret key).Then they respectively compute their public keys: and Here both E and F are points on (6).Then Alice and Bob compute respectively As a result,

Decomposition
Consider randomly selected non-zero integers 1; A  that are co-prime with p. Consider positive integers k, q and r that satisfy ; {for details see Step5 below}; Select integers u, v and w that satisfy Here k and q are secret keys that satisfy where k and q are periodically updated.
There are ten combinations of positive integers satisfying (12); these combinations are listed in lexicographically increasing order in Table 1.Juxtapose  0 0 , 

Numeric Illustration
x y and let G := 8627581549.
Remark2: After t cycles Alice and Bob must use a DHKE to establish a new mutual secret key G.
Choice of A, B and C: one way to choose A and B is to assign even digits of G to A and odd digits of it to B. Then select C that is a multiplicative inverse of AB modulo p: Remark3: The ideas of decomposition can be applied to any secret key; where splitting is completely independent of how this key is established.

Plaintext Pre-conditioning
If there is a pair  

Numeric Illustration Continued
Assign all even digits of G := 8627581549 to A and all odd digits of G to B.
Then A = 67859, {in and then compute encryptors u, v and w (15)-( 17).For {see the 2 nd column in and then compute encryptors u, v and w.
See Table 3 of all encryptors for i = 1,2, t.

Plaintext pre-conditioning
Let = {266, 45769, 37585, 36488, 46572}.Decryption is performed in reverse: since Bob knows the mutual secret key M =  0 0 , x y , he finds A, B, k, q and r; then computes C, R, S, T; and the multiplicative inverse values of u, v and w.

Key Establishment Based on Gaussian Modulus
Consider (l, p) = (1000, 3001); and a generator L = (2269, -2204).All corresponding steps and actions by Alice and Bob are provided in Table 6.gits 1,2, ,9,1,2, into G: Therefore, M = (-0502, 1853) is the mutual secret key established between Alice and Bob.Notice that components in M can be positive and negative.If a component is negative, post digit "2" in front of its left-most digit; if the component is positive, post digit "9" in front of its left-most digit.Therefore M := (20502, 91853).For large l and p in modulus (l, p), the probability is negligibly small that either 0 0 = 0 or 0 x y  .

Conclusion
In the proposed cryptocol it is shown that for every pair of integers in secret key   0 0 , x y there are numerous ways to compute integers {u, v, w} that hide information on the encryption stage.

Acknowledgements
For every digit in juxtaposed G it is possible to encrypt one plaintext array.
With high probability each component in   0 0 , x y 10 log p has the same number of digits t as modulus p. Therefore in G there are about 2t digits.For each digit we select an appropriate combination of keys {k, q, r} from Table 1 and encrypt five blocks of the plaintext.Therefore for every G we can encrypt   5 2 10 10 I express my appreciation to NJIT students A. Koripella and M. Sikorski for their assistance and suggestions that improved style of this paper, to D. Kanevsky for his participation in discussion, and to C. Pomerance and H. Cohen for their advices about complex modulus reduction.
In application, to assure strong crypto-immunity, This is an additional potential for randomization.If a protocol designer of encryption/decryption scheme represents G in a numeric form with base n, then it is possible to select n combinations of secret exponents

References
b, e and p are integer parameters of the EC, and modulus p is an odd real prime[3].If (6) is used for ECDHKE between Alice and Bob, then both parties create a mutual secret key   To preclude this possibility consider the following protocol of plaintext pre-conditioning: subdivide plaintext m into arrays of blocks in such a way that for every block holds example, if s = 4 and n = 16, then we need to specify sixteen combinations of .the number of possible combinations of k's is 35 for d = 8.

Table 2
Since G does not have digits 0 and 3, then only eight of ten combinations of {k, q, r} that are listed in Table1are used to compute the information hiders u, v, w: